Full conglomerability

Abstract

We do a thorough mathematical study of the notion of full conglomerability, that is, conglomerability with respect to all the partitions of an infinite possibility space, in the sense considered by Peter Walley. We consider both the cases of precise and imprecise probability (sets of probabilities). We establish relations between conglomerability and countable additivity, continuity, super-additivity, and marginal extension. Moreover, we discuss the special case where a model is conglomerable with respect to a subset of all the partitions, and try to sort out the different notions of conglomerability present in the literature. We conclude that countable additivity, which is routinely used to impose full conglomerability in the precise case, appears to be the most well-behaved way to do so in the imprecise case as well by taking envelopes of countably additive probabilities. Moreover, we characterize these envelopes by means of a number of necessary and sufficient conditions.